Three Regions
"Our considerations are neither complete nor compelling,
but by explaining a simple idea in its simplest form,
they may form the basis for further speculations."
Karl Schwarzschild (1906)
The perfect fit of the observed ratios to the simplest 1-2-1 (S = 2A) greenhouse model, as depicted by Kevin Trenberth or Dennis Hartmann or Vardavas and Taylor (2007, below) or Marshall and Plumb (2008, below)
and the accuracy of the all-sky extension shows that on a fundamental level Earth's atmosphere follows the ruling principles of that single-slab opaque atmospheric model.
Why and how?
Earth's atmosphere consits of three regions: two infrared-opaque (one by clouds with an effective cloud area fraction of 0.6 and one by GHGs in the clear-sky region); and a transparent part (the window). This geometric arrengement, by the organized interplay of these regions, makes it capable to implement the simplest single-layer radiative transfer principles as depicted by the two clear-sky and two all-sky equations.
In the two IR-opaque regions the simplest model holds, while the window is transparent. Energetic requirements fine-tune the system to satisfy the constraints in the global mean.
The IR-opaque cloudy region
The picture below shows the three regions (with their relative area fractions) and their interplay to maintain the required all-sky system.
(Brown numbers in cloudy units, blue numbers in clear-sky units, red numbers in all-sky units)
OLR is the sum of the cloud-opaque, GHG-opaque and transparent (window) radiations.
Again,
Let us repeat here the six equations in their own units:
Eq. (1) (clear-sky net) 2 = 4/2
Eq. (2) (clear-sky total) 2 + 6 = 2 × 4
Eq. (3) (all-sky net) 4 = (9 – 1) /2
Eq. (4) (all-sky total) 4 + 15 = 2 × 9 + 1
Eq. (3cd) (cloudy net) 2 = (5 – 1)/2
Eq. (4cd) (cloudy total) 2 + 9 = 2 × 5 + 1.
g(cloudy) = (9 – 5)/9 = 4/9
g(clear) = (4 – 2)/6 = 1/3
g(all) = (15 – 9)/15 = 2/5
g(all) = g(cloudy) × βeff + g(clear) × (1 – βeff)
Notice that in the clear-sky region,
g(clear) = g(GHG-opaque) × (5/6) + g(WIN) × (1/6),
but since g(WIN) = 0, it follows that
g(GHG-opaque) = 2/5 = 0.4 = g(all).
The greenhouse factor in the opaque clear-sky region equals to the all-sky value.
The area-weighted upward atmospheric emission in the GHG-opaque clear-sky region equals the all-sky OLR.
As we have seen, the planetary emissivity (the ratio between the LW radiation to space at the upper boundary and the LW radiation from the lower boundary), called also as the all-sky transfer function f(all) = OLR(all)/ULW, according to IPCC AR6 Fig. 7.2 is 239/398 = 0.6005, while its theoretical value is 9/15 = 0.6. Its equivalence to the effective cloud area fraction guarrantees that the upward LW emission from the cloud-covered part of the surface provides the energetic content of the outgoing LW radiation at TOA:
f(all) × ULW = OLR(all) = ULW × βeff,
while the cloud-free region of the surface emiits the LW energy being equal to the greenhous effect:
(1 – f(all)) × ULW = g(all) × ULW = (1 – βeff) × ULW.
This is one of the interplays that establishes the system's stability.